| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 2.31·5-s − 6·6-s + 20.8·7-s − 8·8-s + 9·9-s − 4.62·10-s − 3.68·11-s + 12·12-s − 86.3·13-s − 41.7·14-s + 6.94·15-s + 16·16-s − 18·18-s + 5.54·19-s + 9.25·20-s + 62.6·21-s + 7.37·22-s + 42.5·23-s − 24·24-s − 119.·25-s + 172.·26-s + 27·27-s + 83.4·28-s + 109.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.206·5-s − 0.408·6-s + 1.12·7-s − 0.353·8-s + 0.333·9-s − 0.146·10-s − 0.101·11-s + 0.288·12-s − 1.84·13-s − 0.796·14-s + 0.119·15-s + 0.250·16-s − 0.235·18-s + 0.0669·19-s + 0.103·20-s + 0.650·21-s + 0.0714·22-s + 0.385·23-s − 0.204·24-s − 0.957·25-s + 1.30·26-s + 0.192·27-s + 0.563·28-s + 0.699·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2.31T + 125T^{2} \) |
| 7 | \( 1 - 20.8T + 343T^{2} \) |
| 11 | \( 1 + 3.68T + 1.33e3T^{2} \) |
| 13 | \( 1 + 86.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 5.54T + 6.85e3T^{2} \) |
| 23 | \( 1 - 42.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 109.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 243.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 485.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 69.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 718.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 538.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 506.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 719.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 303.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.407158666728680633400905787577, −7.918168257215368827302206268871, −7.28631348230577837574800889943, −6.37132815182401254086108177191, −5.10927538906281446417719226778, −4.57986980996899387907696847233, −3.13735203400396803637372399991, −2.23816244700810328243993378867, −1.45262260292723218825438112082, 0,
1.45262260292723218825438112082, 2.23816244700810328243993378867, 3.13735203400396803637372399991, 4.57986980996899387907696847233, 5.10927538906281446417719226778, 6.37132815182401254086108177191, 7.28631348230577837574800889943, 7.918168257215368827302206268871, 8.407158666728680633400905787577
